Distributed Computing Graph Drawing - ETH Z 20… · UDG Embedding 2D: Hardness Results 2D NP-hard,...

ETH Zurich – Distributed Computing – www.disco.ethz.ch

Roger Wattenhofer

Graph DrawingDistributed Computing

Zwei Seelen wohnen, ach! in meiner Brust

PODC

SODA

STOCFOCS

ICALPSPAA

EC

SenSysOSDI

Mobicom

Multimedia

Ubicomp

SIGCOMM

HotNets

My Talk: Open Problem

Draw wireless networks modeled by something

by using something to do something.

(plus a few other stories)

Bittorrent Graph[Decker et al., 2013]

But Graph Drawing?

Democrats vs. Republicans Puzzle

Offended by Ugly Graphs….?

Democrats vs. Republicans Lower Bound Graph[Frischknecht et al., 2013]

Nodes Coordinates

MusicGraphs

[Bossard et al., 2009]

Drawing Graphs Without Edges!

“Only Tiny Graphs Should Have Edges!”

Wireless Networks

Which Node is Which?

Position only from Connectivity

Unit Disk Graph (UDG)

UDG Embedding

1D

Easy greedy “Hop-Skip” algorithm [O’Dell et al., 2005]

2D

UDG Embedding 2D: Heuristics

e.g., [Priyanta et al., 2003], [Gotsman et al., 2004], [Bruck et al., 2005], [Kröller et al., 2006]

UDG Embedding 2D: Hardness Results

2D

NP-hard, even with exact distance information [Breu, Kirkpatrick, 1998], or angle information [Aspnes et al., 2004] and [Bruck et al., 2004]. Also APX-hard: [Kuhn et al., 2004]

Approximation? max𝑑no edge with 𝑑edge ≤ 1

Approximation algorithms: First [Moscibroda et al., 2004]

Still best: 𝑂(log2.5 𝑛) approximation [Pemmaraju et al., 2006]

Wireless Beyond UDG

If you gave me $100 for each paper written with

the unit disk assumption, I still could not buy a radio

that is unit disk!

Quasi Unit Disk Graph (QUDG)

Bounded Independence Graph (BIG)

Size of any independent set grows polynomially with hop distance r

Unit Ball Graph (UBG)

Ametric with constant doubling dimension

you only need a constant number of balls of half the radius

Overview Wireless Connectivity Models

too pessimistic too optimistic

GeneralGraph

UDG

QuasiUDG

Bounded Independence

Unit BallGraph

Wireless Beyond Connectivity

InterferenceRange

Protocol Model

ReceptionRange

29

Physical (SINR) Model

From Audio to Wireless

… even with very simple hardware

Time for transmitting 20‘000 packets:

u1 u2 u3 u4 u5 u6

[Moscibroda et al., 2006]

The Power of Power Control

Improving Protocols with Graph Drawing

Possible Application: Hotspots in WLAN

X

Y

Z

Possible Application: Hotspots in WLAN

X

Y

Z

MIMO et al.

The Capture Effect – Power Difference

[König et al., 2016]

Two Transmitters, One Receiver

[König et al., 2016]

Using Distance Information

Trilateration

Disca:Distributed Discussion Statistics

Disca:Distributed Discussion

Statistics

[Bissig et al., 2016]

Distance & Angle?

„People who are really serious about softwareshould make their own hardware.”

Alan Kay

46Ad Hoc and Sensor Networks, Roger

Wattenhofer

SpiderBat

[Oberholzer et al., 2010]

Angle-of-Arrival MeasurementsReceiver Sender

North

East

South

West

Learning Environment with SpiderBat?

SpiderBat: Iterative Art Gallery Problem?

Mother of All NodeCoordinate

Problems

GPS

Audio Radio343 m/s 299 792 458 m/s

GPS Trilateration

Least Squares to Compute Position & Time

Spoofing

Spoofing

Spoofing

Spoofing

How Expensive is Spoofing?

Alternative Approach: Collective Detection

Drawing Routing

Anchor-Based Routing

[W et al., 2005]

Greedy Geometric Routing

[Flury et al., 2009]

1

2

3

9

4

5

6

8

7

1

3

9

8

4

5

6

7

2

Related: self-approaching graph drawings, increasing-chord drawings,

monotone drawings, hyperbolic geometry, …

Summary

1

2

39

4

5

6

8

7

Open Problem

Draw wireless networks modeled by UDG, QUDG, BIG,

UBG by using connectivitiy, interference, distance, angle,

or multipath information to understand which node is which,

or for better protocols (routing, media access).

Thank You!Questions & Comments?

www.disco.ethz.ch